Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(6,8)(10,14)(11,17)(12,19)(22,24)(25,26)(27,30)(28,31)(29,34)(32,38)(33,35)(36,40)(37,39)(41,42,45,48)(43,46,49,53)(44,51,52,47)(50,56,55,54);(1,2,5,4)(3,6,7,8)(10,14)(11,17)(16,21)(18,23)(26,27)(28,35)(32,34)(36,37)(43,49)(46,53)(50,55)(54,56);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,33,35,31)(29,34,38,32)(36,40,37,39)(41,43,45,49)(42,46,48,53)(44,50,52,55)(47,54,51,56);(1,3)(2,6)(4,8)(5,7)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,35)(26,31,27,33)(29,37,38,36)(32,40,34,39)(41,42)(43,46)(44,47)(45,48)(49,53)(50,54)(51,52)(55,56);(1,4,5,2)(3,8,7,6)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)(41,44,45,52)(42,47,48,51)(43,50,49,55)(46,54,53,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=fgh,d^2=g,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4825003);